Average Error: 0.5 → 0.3
Time: 16.5s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \frac{\left(v \cdot v\right) \cdot \frac{5}{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right) - \left(\frac{v \cdot v}{\sqrt{2} \cdot \pi} \cdot \frac{v \cdot v}{t}\right) \cdot \frac{53}{8}\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\left(\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \frac{\left(v \cdot v\right) \cdot \frac{5}{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right) - \left(\frac{v \cdot v}{\sqrt{2} \cdot \pi} \cdot \frac{v \cdot v}{t}\right) \cdot \frac{53}{8}
double f(double v, double t) {
        double r2429451 = 1.0;
        double r2429452 = 5.0;
        double r2429453 = v;
        double r2429454 = r2429453 * r2429453;
        double r2429455 = r2429452 * r2429454;
        double r2429456 = r2429451 - r2429455;
        double r2429457 = atan2(1.0, 0.0);
        double r2429458 = t;
        double r2429459 = r2429457 * r2429458;
        double r2429460 = 2.0;
        double r2429461 = 3.0;
        double r2429462 = r2429461 * r2429454;
        double r2429463 = r2429451 - r2429462;
        double r2429464 = r2429460 * r2429463;
        double r2429465 = sqrt(r2429464);
        double r2429466 = r2429459 * r2429465;
        double r2429467 = r2429451 - r2429454;
        double r2429468 = r2429466 * r2429467;
        double r2429469 = r2429456 / r2429468;
        return r2429469;
}
↓
double f(double v, double t) {
        double r2429470 = 1.0;
        double r2429471 = 2.0;
        double r2429472 = sqrt(r2429471);
        double r2429473 = atan2(1.0, 0.0);
        double r2429474 = r2429472 * r2429473;
        double r2429475 = r2429470 / r2429474;
        double r2429476 = t;
        double r2429477 = r2429475 / r2429476;
        double r2429478 = v;
        double r2429479 = r2429478 * r2429478;
        double r2429480 = 2.5;
        double r2429481 = r2429479 * r2429480;
        double r2429482 = r2429476 * r2429474;
        double r2429483 = r2429481 / r2429482;
        double r2429484 = r2429477 - r2429483;
        double r2429485 = r2429479 / r2429474;
        double r2429486 = r2429479 / r2429476;
        double r2429487 = r2429485 * r2429486;
        double r2429488 = 6.625;
        double r2429489 = r2429487 * r2429488;
        double r2429490 = r2429484 - r2429489;
        return r2429490;
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\frac{53}{8} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)}\]

  3. Simplified0.5

    \[\leadsto \color{blue}{\left(\frac{\frac{1}{t}}{\sqrt{2} \cdot \pi} - \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\right) - \left(\frac{v \cdot v}{\sqrt{2} \cdot \pi} \cdot \frac{v \cdot v}{t}\right) \cdot \frac{53}{8}}\]
  4. Using strategy rm

  5. Applied div-inv0.5

    \[\leadsto \left(\color{blue}{\frac{1}{t} \cdot \frac{1}{\sqrt{2} \cdot \pi}} - \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\right) - \left(\frac{v \cdot v}{\sqrt{2} \cdot \pi} \cdot \frac{v \cdot v}{t}\right) \cdot \frac{53}{8}\]
  6. Using strategy rm

  7. Applied associate-*l/0.3

    \[\leadsto \left(\color{blue}{\frac{1 \cdot \frac{1}{\sqrt{2} \cdot \pi}}{t}} - \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\right) - \left(\frac{v \cdot v}{\sqrt{2} \cdot \pi} \cdot \frac{v \cdot v}{t}\right) \cdot \frac{53}{8}\]

  8. Final simplification0.3

    \[\leadsto \left(\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \frac{\left(v \cdot v\right) \cdot \frac{5}{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right) - \left(\frac{v \cdot v}{\sqrt{2} \cdot \pi} \cdot \frac{v \cdot v}{t}\right) \cdot \frac{53}{8}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))